Abstract
The flows past a sphere and a square cylinder of diameter d moving horizontally at the velocity U in a linearly density-stratified viscous incompressible fluid are studied. The flows are described by the Navier–Stokes equations in the Boussinesq approximation. Variations in the spatial vortex structure of the flows are analyzed in detail in a wide range of dimensionless parameters (such as the Reynolds number Re = Ud/ν and the internal Froude number Fr = U/(Nd), where ν is the kinematic viscosity and N is the buoyancy frequency) by applying mathematical simulation (on supercomputers of Joint Supercomputer Center of the Russian Academy of Sciences) and three-dimensional flow visualization. At 0.005 < Fr < 100, the classification of flow regimes for the sphere (for 1 < Re < 500) and for the cylinder (for 1 < Re < 200) is improved. At Fr = 0 (i.e., at U = 0), the problem of diffusion-induced flow past a sphere leading to the formation of horizontal density layers near the sphere’s upper and lower poles is considered. At Fr = 0.1 and Re = 50, the formation of a steady flow past a square cylinder with wavy hanging density layers in the wake is studied in detail.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. D. Crapper, “A three-dimensional solution for waves in the lee of mountains,” J. Fluid Mech. 6, 51–76 (1959).
Q. Lin, W. R. Lindberg, D. L. Boyer, and H. J. S. Fernando, “Stratified flow past a sphere,” J. Fluid Mech. 240, 315–354 (1992).
J. M. Chomaz, P. Bonneton, and E. J. Hopfinger, “The structure of the near wake of a sphere moving horizontally in a stratified fluid,” J. Fluid Mech. 254, 1–21 (1993).
E. Ya. Sysoeva, Candidate’s Dissertation in Mathematics and Physics (Moscow, 1990).
E. Ya. Sysoeva and Yu. D. Chashechkin, “Vortex structure of a wake behind a sphere in a stratified fluid,” J. Appl. Mech. Tech. Phys. 27 (2), 190–196 (1986).
E. Ya. Sysoeva and Yu. D. Chashechkin, “Spatial structure of a wake behind a sphere in a stratified fluid,” J. Appl. Mech. Tech. Phys. 29 (5), 655–660 (1988).
Yu. D. Chashechkin, “Hydrodynamics of a sphere in a stratified fluid,” Fluid Dyn. 24 (1), 1–7 (1989).
D. L. Boyer, P. A. Davies, H. J. S. Fernando, and X. Zhang, “Linearly stratified flow past a horizontal circular cylinder,” Philos. Trans. R. London Soc. Ser. A: Math. Phys. Sci. 328 (1601), 501–528 (1989).
Yu. D. Chashechkin and I. V. Voeikov, “Vortex systems past a cylinder in a continuously stratified fluid,” Izv. Atmos. Ocean Phys. 29 (6), 821–830 (1993).
V. V. Mitkin and Yu. D. Chashechkin, “Transformation of hanging discontinuities into vortex systems in a stratified flow behind a cylinder,” Fluid Dyn. 42 (1), 12–23 (2007).
H. Hanazaki, “A numerical study of three-dimensional stratified flow past a sphere,” J. Fluid Mech. 192, 393–419 (1988).
P. V. Matyushin, “The vortex structures of the 3D separated stratified fluid flows around a sphere,” Selected Papers of the International Conference on Fluxes and Structures in Fluids 2007 (July 2–5, 2007, Russia), Ed. by Yu. D. Chashechkin and V. G. Baydulov (IPMech. Russ. Acad. Sci., Moscow, 2008), pp. 139–145.
P. V. Matyushin, “Continuous transformation of the vortex structures around a moving sphere with increasing of the fluid stratification,” Selected Papers of the International Conference on Flows and Structures in Fluids: Physics of Geospheres 2009 Ed. by Yu. D. Chashechkin and V. G. Baydulov (IPMech. Russ. Acad. Sci., Moscow, 2008), pp. 251–258.
V. A. Gushchin and P. V. Matyushin, “Numerical simulation and visualization of vortical structure transformation in the flow past a sphere at an increasing degree of stratification,” Comput. Math. Math. Phys. 51 (2), 251–263 (2011).
O. M. Belotserkovskii, V. A. Gushchin, and V. N. Konshin, “The splitting method for investigating flows of a stratified liquid with a free surface,” USSR Comput. Math. Math. Phys. 27 (2), 181–191 (1987).
V. A. Gushchin and V. N. Konshin, “Computational aspects of the splitting method for incompressible flow with a free surface,” J. Comput. Fluids 21 (3), 345–353 (1992).
V. A. Gushchin and P. V. Matyushin, “Numerical simulation of separated flow past a sphere,” Comput. Math. Math. Phys. 37 (9), 1086–1100 (1997).
V. A. Gushchin, A. V. Kostomarov, P. V. Matyushin, and E. R. Pavlyukova, “Direct numerical simulation of the transitional separated fluid flows around a sphere and a circular cylinder,” J. Wind Eng. Ind. Aerodyn. 90 (4–5), 341–358 (2002).
V. A. Gushchin and P. V. Matyushin, “Classification of Regimes of Separated Fluid Flows over a Sphere at Moderate Reynolds Numbers,” in Mathematical Simulation: Problems and Results (Nauka, Moscow, 2003), pp. 199–235 [in Russian].
V. A. Gushchin, A. V. Kostomarov, and P. V. Matyushin, “3D visualization of the separated fluid flows,” J. Visualization 7 (2), 143–150 (2004).
V. A. Gushchin and P. V. Matyushin, “Mathematical modeling of 3D incompressible fluid flows,” Mat. Model. 18 (5), 5–20 (2006).
P. V. Matyushin, Candidate’s Dissertation in Mathematics and Physics (Moscow, 2003).
V. A. Gushchin and P. V. Matyushin, “Vortex formation mechanisms in the wake behind a sphere for 200 < Re < 380,” Fluid Dyn. 41 (5), 795–809 (2006).
L. S. Pontryagin, Ordinary Differential Equations (Addison-Wesley, Reading, Mass., 1962; Nauka, Moscow, 1974).
M. S. Chong, A. E. Perry, and B. J. Cantwell, “A general classification of three-dimensional flow fields,” Phys. Fluids A 2 (5), 765–777 (1990).
J. Jeong and F. Hussain, “On the identification of a vortex,” J. Fluid Mech. 285, 69–94 (1995).
V. G. Baidulov, P. V. Matyushin, and Yu. D. Chashechkin, “Structure of a diffusion-induced flow near a sphere in a continuously stratified fluid,” Dokl. Phys. 50, 195–199 (2005).
V. G. Baidulov, P. V. Matyushin, and Yu. D. Chashechkin, “Evolution of the diffusion-induced flow over a sphere submerged in a continuously stratified fluid,” Fluid Dyn. 42 (2), 255–267 (2007).
K. E. B. Lofquist and L. P. Purtell, “Drag on a sphere moving horizontally through a stratified liquid,” J. Fluid Mech. 148, 271–284 (1984).
P. J. Mason, “Forces on spheres moving horizontally in a rotating stratified fluid,” Geophys. Astrophys. Fluid Dyn. 8, 137–154 (1977).
P. A. Sheppard, “Air flow over mountains,” Q. J. R. Met. Soc. 82, 528–529 (1956).
P. V. Matyushin, “Classification of the regimes of the stratified viscous fluid flows around a 2D square cylinder,” Proceedings of the International Conference on Fluxes and Structures in Fluids (June 25–28, 2013, Russian State Hydrometeorological University, St. Petersburg) (Maks-Press, Moscow, 2013), pp. 209–212.
C. Norberg, A. Sohankar, and L. Davidson, “Numerical simulation of unsteady flows around a square twodimensional cylinder,” Twelfth Australian Fluid Mechanics Conference (Univ. of Sydney, Australia, Sydney, 1995), pp. 517–520.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of O.M. Belotserkovskii
Original Russian Text © V.A. Gushchin, P.V. Matyushin, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 6, pp. 1049–1063.
Rights and permissions
About this article
Cite this article
Gushchin, V.A., Matyushin, P.V. Simulation and study of stratified flows around finite bodies. Comput. Math. and Math. Phys. 56, 1034–1047 (2016). https://doi.org/10.1134/S0965542516060142
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542516060142